Question

Prove that the diagonals of a rhombus are perpendicular using coordinate geometry

Answer 1

If we draw a rhombus OABC on a coordinate grid with one vertex at the point O (0,0)

and let its height be 1 then (using the Pythagoras theorem) the coordinates of the 4 points will be as follows:-

O = (0,0) , A = (1, 1), B = (sqrt2+ 1 , 1) , C = (sqrt2, 0)

Slope of the diagonal OB = 1 / (sqrt2 + 1)

Slope of the other diagonal AC = - 1 / (sqrt2 - 1)

Product of the slopes = - 1 / ( sqrt2 + 1)(sqrt2 - 1) = -1 / 1 = -1

This proves that the diagonals are perpendicular.